Calculus Examples

$\int \frac{\sec (x)}{\tan^{2} (x)} d x$

Step 1

Simplify.

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Step 1.1

Rewrite $\sec (x)$ as $\frac{\csc (x)}{\cot (x)}$ .

$\int \frac{\frac{\csc (x)}{\cot (x)}}{\tan^{2} (x)} d x$

Step 1.2

Apply the reciprocal identity.

$\int \frac{\frac{\csc (x)}{\cot (x)}}{{(\frac{1}{\cot (x)})}^{2}} d x$

Step 1.3

Simplify.

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Step 1.3.1

Multiply the numerator by the reciprocal of the denominator.

$\int \frac{\csc (x)}{\cot (x)} \cdot \frac{1}{{(\frac{1}{\cot (x)})}^{2}} d x$

Step 1.3.2

Combine.

$\int \frac{\csc (x) \cdot 1}{\cot (x) {(\frac{1}{\cot (x)})}^{2}} d x$

Step 1.3.3

Multiply $\csc (x)$ by $1$ .

$\int \frac{\csc (x)}{\cot (x) {(\frac{1}{\cot (x)})}^{2}} d x$

Step 1.3.4

Simplify the denominator.

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Step 1.3.4.1

Apply the product rule to $\frac{1}{\cot (x)}$ .

$\int \frac{\csc (x)}{\cot (x) \frac{1^{2}}{{\cot (x)}^{2}}} d x$

Step 1.3.4.2

One to any power is one.

$\int \frac{\csc (x)}{\cot (x) \frac{1}{{\cot (x)}^{2}}} d x$

Step 1.3.5

Combine $\cot (x)$ and $\frac{1}{{\cot (x)}^{2}}$ .

$\int \frac{\csc (x)}{\frac{\cot (x)}{{\cot (x)}^{2}}} d x$

Step 1.3.6

Reduce the expression $\frac{\cot (x)}{{\cot (x)}^{2}}$ by cancelling the common factors.

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Step 1.3.6.1

Multiply by $1$ .

$\int \frac{\csc (x)}{\frac{\cot (x) \cdot 1}{{\cot (x)}^{2}}} d x$

Step 1.3.6.2

Factor $\cot (x)$ out of ${\cot (x)}^{2}$ .

$\int \frac{\csc (x)}{\frac{\cot (x) \cdot 1}{\cot (x) \cot (x)}} d x$

Step 1.3.6.3

Cancel the common factor.

$\int \frac{\csc (x)}{\frac{\cot (x) \cdot 1}{\cot (x) \cot (x)}} d x$

Step 1.3.6.4

Rewrite the expression.

$\int \frac{\csc (x)}{\frac{1}{\cot (x)}} d x$

Step 1.3.7

Multiply the numerator by the reciprocal of the denominator.

$\int \csc (x) \cot (x) d x$

Step 2

Since the derivative of $- \csc (x)$ is $\csc (x) \cot (x)$ , the integral of $\csc (x) \cot (x)$ is $- \csc (x)$ .

$- \csc (x) + C$